On Functional Decomposition of Multivariate Polynomials with Differentiation and Homogenization

In this paper, we give a theoretical analysis for the algorithms to compute functional decomposition for multivariate polynomials based on differentiation and homogenization which are proposed by Ye, Dai, Lam (1999) and Faug`ere, Perret (2006, 2008, 2009). We prove a conjecture proposed by Ye, Dai, and Lam (1999) on recovering quadratic forms from the linear combination. We also show that the decomposition for a set of polynomials can be computed from that of its homogenization with high probability. Finally, we prove that the right decomposition factors for a set of polynomials can be computed from its right decomposition factor space. Combining these results together, we prove that the algorithm can compute a degree proper decomposition for a set of randomly decomposable quartic polynomials with probability one when the base field is of characteristic zero, and with probability close to one when the base field is a finite field with sufficiently large odd number. Keywords. Functional...

• Type: Journal
• Year: 2010
• Where: CORR
• Authors: Shang-Wei Zhao, Ruyong Feng, Xiao-Shan Gao
• Keywords: CORR 2010, Decomposition, Education, Polynomials, Right Decomposition Factor